Solving Systems Of Equations ½x - ⅓y = -1 And ⅔x + ½y = 1 ½ A Step-by-Step Guide

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Hey guys! Ever found yourself staring blankly at a system of equations, feeling like you're trying to decipher an ancient scroll? Well, you're not alone! Math problems, especially those involving fractions, can seem daunting at first. But don't worry, we're here to break down the process of solving the system of equations Tolong aku½x - ⅓y = -1 and ⅔x + ½y = 1 ½ into manageable steps. We'll walk through it together, so by the end of this guide, you'll feel confident tackling similar problems. So, grab your pencils and let's dive in! Remember, the key to mastering math is understanding the fundamentals and practicing consistently. Think of it like learning a new language; the more you practice, the more fluent you become. And who knows, you might even start to enjoy the challenge! This journey through algebraic solutions will not only enhance your problem-solving skills but also provide a solid foundation for more advanced mathematical concepts. So, let's embark on this mathematical adventure together and unravel the mysteries of simultaneous equations!

Understanding the Problem: Equations with Fractions

So, the first thing we need to do is understand what we're dealing with. We have two equations:

  1. Tolong aku½x - ⅓y = -1
  2. ⅔x + ½y = 1 ½

These are called simultaneous equations, or a system of equations. Basically, we're looking for values of 'x' and 'y' that make both equations true at the same time. The tricky part here is those pesky fractions! Fractions can sometimes make things look more complicated than they actually are. But don't sweat it, we're going to eliminate them. Our main goal in this section is to really grasp what this system of equations represents. We'll dissect each equation, identify the unknowns (which are 'x' and 'y' in this case), and highlight the fractional coefficients that we need to address. By breaking down the problem into smaller, more digestible chunks, we can pave the way for a smoother solving process. We'll also touch upon the underlying concept of simultaneous equations and why finding a solution that satisfies both equations is crucial. This initial understanding is the bedrock upon which we'll build our solution strategy. So, let's put on our detective hats and begin our investigation into the world of fractions and equations! Remember, every complex problem can be simplified with the right approach, and we're here to guide you every step of the way.

Clearing the Fractions: A Simple Transformation

The best way to deal with fractions in equations is to get rid of them! We can do this by multiplying both sides of each equation by the least common multiple (LCM) of the denominators. Let's start with the first equation: Tolong aku½x - ⅓y = -1. The denominators are 2 and 3. The LCM of 2 and 3 is 6. So, we multiply both sides of the equation by 6:

6 * (Tolong aku½x - ⅓y) = 6 * (-1)

This simplifies to:

3x - 2y = -6

Now, let's do the same for the second equation: ⅔x + ½y = 1 ½. First, we need to convert 1 ½ to an improper fraction, which is 3/2. So our equation becomes ⅔x + ½y = 3/2. The denominators are 3 and 2, and their LCM is 6. We multiply both sides by 6:

6 * (⅔x + ½y) = 6 * (3/2)

This simplifies to:

4x + 3y = 9

Now we have two new equations that are much easier to work with:

  1. 3x - 2y = -6
  2. 4x + 3y = 9

This step is crucial in simplifying the entire solving process. By eliminating fractions, we transform the original equations into a more manageable form. Think of it as decluttering your workspace before starting a project; it makes the task at hand less overwhelming and more approachable. We've essentially cleared the obstacles that were hindering our progress and laid a solid foundation for the next steps. This technique of multiplying by the LCM is a powerful tool in algebra and will come in handy in many different situations. So, make sure you've got this step down pat! We're one step closer to solving this system of equations, and with each step, we're building our confidence and mathematical prowess.

Choosing a Method: Elimination or Substitution

Now that we have our simplified equations, we can choose a method to solve them. The two most common methods are elimination and substitution. Let's briefly discuss each:

  • Elimination: This method involves manipulating the equations so that either the 'x' or 'y' coefficients are opposites. Then, we add the equations together, which eliminates one variable, allowing us to solve for the other. This method is often preferred when the coefficients of one variable are easily made opposites.
  • Substitution: This method involves solving one equation for one variable (e.g., solving for 'x' in terms of 'y') and then substituting that expression into the other equation. This eliminates one variable, allowing us to solve for the other. Substitution is particularly useful when one equation can be easily solved for one variable.

For this particular system of equations, the elimination method seems like a good fit. We can easily manipulate the equations to eliminate either 'x' or 'y'. The key here is to strategically multiply the equations so that the coefficients of one variable become opposites. This decision-making process is a crucial aspect of problem-solving in mathematics. It's not just about getting the right answer; it's about choosing the most efficient and effective path to the solution. We're not just blindly applying formulas; we're actively thinking about the structure of the equations and making informed choices. This is what separates a good problem-solver from a great one. So, let's embrace this opportunity to strategize and choose the method that will lead us to the solution with the least amount of effort and the most clarity. Remember, there's often more than one way to skin a cat, and in this case, we're choosing the method that best suits our needs.

Elimination in Action: Solving for Y

Let's use elimination to solve for 'y' first. We have the equations:

  1. 3x - 2y = -6
  2. 4x + 3y = 9

To eliminate 'x', we need to make the coefficients of 'x' opposites. We can do this by multiplying the first equation by 4 and the second equation by -3:

  • (3x - 2y = -6) * 4 -> 12x - 8y = -24
  • (4x + 3y = 9) * -3 -> -12x - 9y = -27

Now, we add the two equations together:

12x - 8y = -24

  • -12x - 9y = -27
  -17y = -51

Divide both sides by -17:

y = 3

Great! We've found the value of 'y'. This step demonstrates the power of the elimination method. By strategically manipulating the equations, we were able to isolate 'y' and determine its value. It's like a mathematical magic trick, where we make a variable disappear, leaving us with a much simpler equation to solve. But it's not magic; it's the beauty of algebraic manipulation. We're applying the rules of algebra to transform the equations and reveal the hidden solution. This process also highlights the importance of careful attention to detail. One small mistake in multiplication or addition can throw off the entire calculation. So, it's crucial to double-check our work at each step to ensure accuracy. With 'y' solved, we're halfway there. Now, we just need to find the value of 'x', and we'll have the complete solution to our system of equations. The momentum is building, and we're on the verge of a mathematical triumph!

Finding X: Substituting Y Back In

Now that we know y = 3, we can substitute this value into either of our simplified equations to solve for 'x'. Let's use the first equation, 3x - 2y = -6:

3x - 2(3) = -6

3x - 6 = -6

Add 6 to both sides:

3x = 0

Divide both sides by 3:

x = 0

So, we've found that x = 0. This step brings us full circle in the solving process. We've gone from a complex system of equations with fractions to a clear and concise solution. By substituting the value of 'y' back into one of the equations, we were able to unlock the value of 'x'. This highlights the interconnectedness of the variables in a system of equations. Each variable plays a role in determining the value of the others. This process also underscores the elegance of mathematics. We're using a series of logical steps to arrive at a definitive answer. There's a certain satisfaction in knowing that we've solved the puzzle and found the values that make both equations true. With both 'x' and 'y' in hand, we can confidently say that we've cracked the code of this system of equations. But before we celebrate, it's always a good idea to verify our solution to ensure that we haven't made any mistakes along the way.

Verification: Ensuring the Solution is Correct

To make sure our solution (x = 0, y = 3) is correct, we need to substitute these values back into the original equations:

  1. Tolong aku½x - ⅓y = -1

    (½ * 0) - (⅓ * 3) = -1

    0 - 1 = -1

    -1 = -1 (This is true!)

  2. ⅔x + ½y = 1 ½

    (⅔ * 0) + (½ * 3) = 3/2

    0 + 3/2 = 3/2

    3/2 = 3/2 (This is also true!)

Since our solution satisfies both equations, we know we've got it right! This verification step is the final seal of approval on our solution. It's like the quality control check at the end of a production line, ensuring that the product meets the required standards. In this case, our product is the solution to the system of equations, and the standard is that it must satisfy both equations. By substituting our values for 'x' and 'y' back into the original equations, we've confirmed that our solution holds true. This not only gives us confidence in our answer but also reinforces the importance of accuracy in mathematical problem-solving. A small mistake in the earlier steps can lead to an incorrect solution, which is why verification is such a crucial step. With the verification complete, we can now confidently declare victory! We've successfully navigated the world of simultaneous equations and emerged with the correct solution.

Final Answer: The Solution Set

The solution to the system of equations is x = 0 and y = 3. We can write this as an ordered pair: (0, 3). This ordered pair represents the point where the two lines represented by the equations intersect on a graph. This final answer is the culmination of all our efforts. It's the destination we've been working towards since the beginning of the problem. The ordered pair (0, 3) represents the unique point that satisfies both equations in the system. It's the intersection of the two lines, the place where they meet and agree. This solution is not just a pair of numbers; it's a representation of the relationship between the two equations. It's a concise and elegant way to express the values of 'x' and 'y' that make both statements true. And it's a testament to our problem-solving skills and our ability to navigate the world of algebra. So, let's take a moment to appreciate the beauty of this solution and the journey we took to arrive at it. We've not only solved a system of equations, but we've also reinforced our understanding of mathematical concepts and our ability to tackle challenging problems. Congratulations, we did it!

Practice Makes Perfect: Further Exploration

Now that you've seen how to solve this system of equations, try tackling similar problems on your own! The more you practice, the more comfortable you'll become with the process. Remember, math is like a muscle; the more you use it, the stronger it gets. And don't be afraid to make mistakes; they're a natural part of the learning process. The key is to learn from your mistakes and keep practicing. You can find plenty of practice problems online or in textbooks. You can also try creating your own systems of equations and challenging yourself to solve them. The possibilities are endless! So, go forth and conquer the world of mathematics. You've got the tools, you've got the knowledge, and you've got the determination. And remember, we're here to support you every step of the way. So, keep practicing, keep exploring, and keep having fun with math!